Download PRACTICE TEST #4 – FULL ANALYSIS Topics to Know Explanation

Topics to Know
PRACTICE TEST #4 – FULL ANALYSIS
Explanation
1.
How to factor
2.
Inverse Functions
Switching logs to exponents
(𝑙𝑜𝑔𝑏 𝑎 = 𝑒 ↔ 𝑏 𝑒 = 𝑎)
Multiplying binomials (FOIL)
Write it out as two binomials first, and FOIL
Converting imaginary numbers carefully. Remember that no ‘powers’ of 𝑖
can be in your final answer. 𝑖 2 = −1
3.
4.
5.
6.
7.
8.
How to work with complex
fractions
GCF always goes first, if possible.
Then, case I or case II trinomial factoring
To take the inverse of a function, (1) switch
x and y, then (2) re-solve for y
Two steps (1) find the LCD of all little
fractions, (2) multiply each piece by the
FULL LCD and cancel what you can. The
point is to get all of the ‘little’ fractions to
cancel out.
Solving inequalities and testing Set equal to 9 to find your critical points.
Think what’s inside the absolute value
brackets can be equal to 9 (because the
absolute value of 9 is 9), or it could be equal
to -9 (because the absolute value of -9 is 9)
Once you have the critical points, test any
number you’d like in each of the three
sections back into the original inequality to
see what sections are true, and which are
false
Calculator Skills!
Enter the data into List1
Run 1-Var Stats
Sample standard deviation is 𝑺𝑥
Domain Restrictions
Domain is all x-values. You can plug in any
Denominators ≠ 0
real number you can think of for x, unless it
Radicals ≥ 0
makes this denominator equal zero… that’s
a no-no
Composition of functions
Plug 5 into the f function first. Then plug
Work from the inside out
that answer into the g function
Work
2
5𝑥(𝑥 − 4𝑥 + 3) = 0
5𝑥(𝑥 − 3)(𝑥 − 1) = 0
𝑦 = 𝑙𝑜𝑔3 𝑥
𝑥 = 𝑙𝑜𝑔3 𝑦 (switch x and y)
3𝑥 = 𝑦 (convert log into exponent to re-solve for y)
(3 − 5𝑖)(3 − 5𝑖)
9 − 15𝑖 − 15𝑖 + 25𝑖 2
9 − 30𝑖 + 25𝑖 2
9 − 30𝑖 − 25
−16 − 30𝑖
1
+1
𝑥
1
−1
𝑥
LCD: x
1
(𝑥) + (𝑥)1
𝑥
1
(𝑥) − (𝑥)1
𝑥
1+𝑥
1−𝑥
2𝑥 + 3 = 9 2𝑥 + 3 = −9
𝑥=3
𝑥 = −6
Test:
𝑥 = −8
𝑥=0
|2(−8) + 3| < 9
|2(0) + 3| < 9
13 < 9
3<9
False
TRUE
2𝑥 + 10 ≠ 0
𝑥 ≠ −5
𝑓(5) = 4(5) − 5 = 15
𝑔(15) = −2(15) + 7 = −𝟐𝟑
𝑥=4
|2(4) + 3| < 9
11 < 9
False
PRACTICE TEST #4 – FULL ANALYSIS
9.
Simplifying Radicals
Break each individual radical down. Terms
can be combined when the variables and the
radicals match
10.
Function: No x-values repeat
One-to-one: No y-values repeat
11.
(Reference sheet formula)
Look for an answer choice that does not
have any repeating y’s (one-to-one) and
does not have any repeating x’s (function)
Draw & Label. Ensure that the angle used is
in between the two sides used.
12.
13.
14.
Function: No x-values repeat
Calculator skills!
Multiplying Binomials
Pythagorean Trig Identities
15.
Shifts
16.
Sum & Product formulas
17.
(Reference sheet formula)
18.
19.
4𝑥√4√5𝑥 − 5√𝑥 2 √5𝑥
4𝑥 ⋅ 2√5𝑥 − 5 ⋅ 𝑥√5𝑥
8𝑥√5𝑥 − 5𝑥√5𝑥
3𝑥√5𝑥
1
𝐴𝑟𝑒𝑎 = 2 (23)(14)(sin 71)
𝐴𝑟𝑒𝑎 = 152.2
Vertical line test
Multiply them out carefully
Then, using Pythagorean identities (you
know it’s pythag because there’s a trig
function squared), turn it into one trig
function
The number added/subtracted attached to
the x is your left & right movement (it’s
backwards)
The number added/subtracted outside of
the x is your up & down movement
𝑏
𝑐
Sum = − 𝑎
Product = 𝑎
(1 − cos 𝑥)(1 + cos 𝑥)
1 − cos2 𝑥
Pythag Identity:
sin2 𝑥 + cos 2 𝑥 = 1
sin2 𝑥 = 1 − cos 2 𝑥
sin2 𝑥
−(−4)
1
9
= 41
Sum =
Use a quick right triangle & pythag theorem
𝑂𝑝𝑝
to find the value of sin (𝐻𝑦𝑝)
cos 𝑥
Switch from log to exponent
Variable stuck in a log… switch to exponent
Arc Length formula
Remember the angle must be in radians (it
usually is)
51 = 𝑥 + 4
5=𝑥+4
𝑥=1
𝑠
𝜃=𝑟
=4
8
1
Product = = 8
40
sin 𝑥 = 41
sin 2𝑥 = 2(sin 𝑥)(cos 𝑥)
40
9
720
sin 2𝑥 = 2 (41) (41) = 1681
𝑠
2 = 10
𝑠 = 20
PRACTICE TEST #4 – FULL ANALYSIS
20.
Review packet, Trig page,
bottom left corner… SO
IMPORTANT!!!
Isolate (this one is simple solving)… ref
angle… quadrants… put it all together
Isolate: cos 𝜃 = 2
(1) 𝜃 = 45°
(2) Quads 2 & 3
(3) 45° in quad 2 = 135°
45° in quad 3 = 225°
21.
Axis of symmetry formula
𝑏
𝑥 = − 2𝑎 (the non-radical part
of the quadratic formula!)
𝑥=
22.
Factorial Formulas
Watch out for double-negatives.
Also can graph the parabola in your
calculator and look for where the symmetry
is
The numerator is the total number of
letters, the denominator backs out the
repeats
Isolate the absolute value part
Think what’s inside the absolute value
brackets can be equal to (because the
absolute value of 8 is 8), or it could be equal
to -8 (because the absolute value of -8 is 8)
Sine goes through origin; cosine doesn’t
Amplitude tells you how high the graph goes
Frequency tells you how many FULL curves
happen in 360°
Also... plug the options into your calculator
and see which one matches! (Zoom 7: ZTrig)
For the 𝑟 𝑡ℎ term of (𝑎 + 𝑏)𝑛
( 𝑛 𝐶𝑟−1 )(𝑎𝑛−(𝑟−1) )(𝑏𝑟−1 )
|3𝑥 − 4| = 8
3𝑥 − 4 = 8
𝑥=4
Move all of the negative exponents to the
other side of the fraction line first, then
combine and/or simplify as much as
possible
Look for factors that multiply to -24 (-8*3)
and add to -10. Rewrite as 4 terms, then
factor by grouping.
Isolate the radical first
Square both sides to get rid of the radical
20𝑥 5
35𝑥 3 𝑦 2 𝑦 3
20𝑥 5
4𝑥 2
=
35𝑥 3 𝑦 5 7𝑦 5
23.
Solving Inequalities
24.
Graphs of trig functions
Amplitude
Frequency
25.
Binomial Expansion Formula
Just memorize it... I promise you
want to… it’s a regent’s favorite
26.
Negative exponents
27.
Case II Trinomial Factoring
28.
Solving Radical Expressions
−√2
−(−8)
2(4)
8
=8=1
Total: 10 letters
A: 2 R: 2
10!
2!2!
3𝑥 − 4 = −8
4
𝑥 = −3
𝑛 = 6, 𝑟 = 4, 𝑟 − 1 = 3
( 6 𝐶3 )(𝑥 6−3 )(43 )
(20)(𝑥 3 )(64)
1280𝑥 3
3𝑥 2 − 12𝑥 + 2𝑥 − 8
3𝑥(𝑥 − 4) + 2(𝑥 − 4)
(3𝑥 + 2)(𝑥 − 4)
√3𝑥 − 5 = 2
3𝑥 − 5 = 4
𝑥=3
PRACTICE TEST #4 – FULL ANALYSIS
29.
30.
Compliments of Cofunctions
are Equal
THE TWO ANGLES ARE NOT
EQUAL
(Reference Sheet Formula)
Sin and cos are COfunctions
They are equal
SO, the two angles must be compliments
(add to 90°)
2 sides 2 angles… Law of SINES
31.
Circle Formula
32.
Solving Rational Expressions
Center... opposite sign
Radius… square root
Find LCD
Multiply each term by what’s MISSING
Solve
3𝑥 − 4 + 5𝑥 + 14 = 90
8𝑥 + 10 = 90
𝑥 = 10
23
sin 61
𝑏
= sin 40
𝑏 = 16.9
Center (-15, -1)
Radius = √196 = 14
10
4
2
+ =
LCD: (𝑥 − 3)(3)
𝑥−3
3
𝑥−3
3 10
𝑥−3 4
3
2
(3) 𝑥−3 + (𝑥−3) 3 = (3) 𝑥−3
30
4𝑥−12
6
+ 3(𝑥−3) = 3(𝑥−3)
3(𝑥−3)
4𝑥+18
6
= 3(𝑥−3)
3(𝑥−3)
4𝑥 + 18 = 6
𝑥 = −3
33.
Rationalizing Denominators
Multiply top and bottom by the conjugate.
Simplify carefully
34.
Sigma
Plug in all integers from the bottom number
(4) to the top numbers (7) into the
expression given. Add up all of the answers
35.
Plug in
𝑒 is a number, not a variable
Quadratic Formula
36a
Carefully go through the arithmetic &
simplifying
26
7−√5
⋅
7+√5 7−√5
182−26√5 182−26√5 2(91−13√5) 91−13√5
=
=
= 22
49−5
44
44
𝑥
𝑥
𝑥
𝑥
=4
=5
=6
=7
5(4) − 12 = 8
5(5) − 12 = 13
5(6) − 12 = 18
5(7) − 12 = 23
8 + 13 + 18 + 23 = 62
𝑉 = (5200)(𝑒 0.041∗12 )
$8,505.04
−(−6) ± √(−6)2 − 4(1)(34)
𝑥=
2(1)
6 ± √36 − 136
𝑥=
2
6 ± √−100
𝑥=
2
6 ± 10𝑖
𝑥=
2
𝑥 = 3 ± 5𝑖
PRACTICE TEST #4 – FULL ANALYSIS
36b Factor by Grouping
Split into two groups of two, GCF Factor
each group. The leftover binomial should
match!
37a
Input to L1 and L2. Stat Calc LinReg
Calculator Statistics
37b Binomial Probability
38a
Magnitude setup
(parallelogram & triangle)
Reference Sheet Formula
38b Reference Sheet Formula
( 𝑛 𝐶𝑟 )(𝑝𝑟 )(𝑞 𝑛−𝑟 )
n=total, r=want, p=prob of success, q=prob
of failure
Success raised to success.. failure raised to
failure
Setup the parallelogram first, consecutive
angles are supplementary. Label everything
you know. You’re solving for the diagonal
through the middle.
Pull out & relabel a triangle
3 sides, 1 angle… Law of COSINES
2 sides 2 angles… Law of SINES
2𝑥 3 − 10𝑥 2 − 18𝑥 + 90 = 0
2𝑥 2 (𝑥 − 5) − 18(𝑥 − 5) = 0
(2𝑥 2 − 18) (𝑥 − 5) = 0
2𝑥 2 − 18 = 0
𝑥−5=0
𝑥 = −3 𝑥 = 3 𝑥 = 5
a) 𝑦 = 0.774𝑥 + 19.093
b) 𝑟 = 0.9045 (Make sure diagnostics are ON)
c) 𝑦 = 0.774(12) + 19.093
𝑦 = $28
“At least 5” means 5 OR 6
5 games won: ( 6 𝐶5 )(. 635 )(. 371 ) = 0.2203
6 games won: ( 6 𝐶6 )(. 636 )(. 370 ) = 0.0625
“OR” means add them together… 0.2828
𝑥 2 = 422 + 652 − 2(42)(65)(cos 140)
16.4
sin 68
12.7
sin 𝑄
12.7∗sin 68
= 16.4
−1
=
sin 𝑄
𝑄 = sin
39
Review packet, Trig page,
bottom left corner… SO
IMPORTANT!!!
Isolate (this one is caseI factoring)… ref
angle… quadrants… put it all together
= 0.7180
0.7180 = 46°
tan2 𝜃 − 4 tan 𝜃 − 12 = 0
(tan 𝜃 − 6)(tan 𝜃 + 2) = 0
tan 𝜃 − 6 = 0
tan 𝜃 = 6
(1) 𝜃 = tan−1 6
𝜃 = 81°
(2) Positive. Quads 1 & 3
(3) 81° in quad 1 = 81°
81° in quad 3 = 261°
tan 𝜃 + 2 = 0
tan 𝜃 = −2
(1) 𝜃 = tan−1 2
𝜃 = 63°
(2) Negative. Quads 2 & 4
(3) 63° in quad 2 = 117°
63° in quad 4 = 297°
𝜽 = {𝟖𝟏°, 𝟏𝟏𝟕°, 𝟐𝟔𝟏°, 𝟐𝟗𝟕°}